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2014

Bidirectional Control of Absence Seizures by the Basal Ganglia:

A Computational Evidence

Mingming Chen1,∗, Daqing Guo1,∗,†, Tiebin Wang1, Wei Jing1, Yang Xia1,

Peng Xu1, Cheng Luo1, Pedro A. Valdes-Sosa1,2, and Dezhong Yao1,†

1Key Laboratory for NeuroInformation of Ministry of Education, School of Life Science and Technology,

University of Electronic Science and Technology of China, Chengdu 610054, People’s Republic of China2Cuban Neuroscience Center, Ave 25 #15202 esquina 158, Cubanacan, Playa, Cuba

January 13, 2014

Abstract

Absence epilepsy is believed to be associated with the abnormal interactions between the cerebral

cortex and thalamus. Besides the direct coupling, anatomical evidence indicates that the cerebral

cortex and thalamus also communicate indirectly through an important intermediate bridge–basal

ganglia. It has been thus postulated that the basal ganglia might play key roles in the modulation

of absence seizures, but the relevant biophysical mechanisms are still not completely established.

Using a biophysically based model, we demonstrate here that the typical absence seizure activities

can be controlled and modulated by the direct GABAergic projections from the substantia nigra pars

reticulata (SNr) to either the thalamic reticular nucleus (TRN) or the specific relay nuclei (SRN) of

thalamus, through different biophysical mechanisms. Under certain conditions, these two types of

seizure control are observed to coexist in the same network. More importantly, due to the competition

between the inhibitory SNr-TRN and SNr-SRN pathways, we find that both decreasing and increasing

the activation of SNr neurons from the normal level may considerably suppress the generation of

SWDs in the coexistence region. Overall, these results highlight the bidirectional functional roles of

basal ganglia in controlling and modulating absence seizures, and might provide novel insights into

the therapeutic treatments of this brain disorder.

Author Summary

Epilepsy is a general term for conditions with recurring seizures. Absence seizures are one of several

kinds of seizures, which are characterized by typical 2-4 Hz spike-and-slow wave discharges (SWDs).

There is accumulating evidence that absence seizures are due to abnormal interactions between cerebral

cortex and thalamus, and the basal ganglia may take part in controlling such brain disease via the indirect

basal ganglia-thalamic pathway relaying at superior colliculus. Actually, the basal ganglia not only send

indirect signals to thalamus, but also communicate with several key nuclei of thalamus through multiple

direct GABAergic projections. Nevertheless, whether and how these direct pathways regulate absence

seizure activities are still remain unknown. By computational modelling, we predicted that two direct

inhibitory basal ganglia-thalamic pathways emitting from the substantia nigra pars reticulata may also

participate in the control of absence seizures. Furthermore, we showed that these two types of seizure

∗These authors contributed to this work equally†Corresponding authors: DY: dyao@uestc.edu.cn; DG: dqguo@uestc.edu.cn. Tel: +86-028-83201018, Fax: +86-028-

83208238, Address: #4, Section 2, North JianShe Road, Chengdu 610054, People’s Republic of China.

1

control can coexist in the same network, and depending on the instant network state, both lowing and

increasing the activation of SNr neurons may inhibit the SWDs due to the existence of competition. Our

findings emphasize the bidirectional modulation effects of basal ganglia on absence seizures, and might

have physiological implications on the treatment of absence epilepsy.

Introduction

Absence epilepsy is a generalized non-convulsive seizure disorder of the brain, mainly occurring in the

childhood years [1]. A typical attack of absence seizures is characterized by a brief loss of consciousness

that starts and terminates abruptly, and meanwhile an electrophysiological hallmark, i.e. the bilaterally

synchronous spike and wave discharges (SWDs) with a slow frequency at approximately 2-4 Hz, can be

observed on the electroencephalogram (EEG) of patients [1, 2]. There is a broad consensus that the

generation of SWDs during absence seizures is due to the abnormal interactions between cerebral cortex

and thalamus, which together form the so-called corticothalamic system. The direct evidence in support

of this view is based on simultaneous recordings of cortex and thalamus from both rodent animal models

and clinical patients [3–5]. Recent computational modelling studies on this prominent brain disorder also

approved the above viewpoint and provided more deep insights into the possible generation mechanism

of SWDs in the corticothalamic system [6–13].

The basal ganglia comprise a group of interconnected subcortical nucleus and, as a whole, represent

one fundamental processing unit of the brain. It has been reported that the basal ganglia are highly

associated with a variety of brain functions and diseases, such as cognitive [14], emotional functions [15],

motor control [16], Parkinson’s disease [17, 18], and epilepsy [19, 20]. Anatomically, the basal ganglia

receive multiple projections from both the cerebral cortex and thalamus, and in turn send both direct and

indirect output projections to the thalamus. These connections enable the activities of the basal ganglia

to influence the dynamics of the corticothalamic system. Therefore, it is naturally expected that the

basal ganglia may provide an active role in mediating between seizure and non-seizure states for absence

epileptic patients. Such hypothesis has been confirmed by both previous animal experiments [19, 21–

23] and recent human neuroimage data [20, 25, 26]. Nevertheless, due to the complicated interactions

between basal ganglia and thalamus, the underlying neural mechanisms on how the basal ganglia control

the absence seizure activities are still remain unclear.

From the anatomical perspective, the substantia nigra pars reticulata (SNr) is one of the major

output nucleus of the basal ganglia to thalamus. Previous experimental studies using various rodent

animal models have demonstrated that suitable changes in the firing of SNr neurons can modulate the

occurrence of absence seizures [21–24]. Specifically, it has been found that pharmacological inactivation of

the SNr by injecting γ-aminobutyric acids (GABA) agonists or glutamate antagonists suppresses absence

seizures[21, 22]. Such antiepileptic effect was supposed to be attributed to the overall inhibitory effect

of the indirect pathway from the SNr to thalamic reticular nucleus (TRN) relaying at superior colliculus

[21, 22]. In addition to this indirect inhibitory pathway, it is known that the SNr also contains GABAergic

neurons directly projecting to the TRN and specific relay nuclei (SRN) of thalamus [27, 28]. Theoretically,

changing the activation level of SNr may also significantly impact the firing activities of SRN and TRN

neurons [28, 29]. This contribution might further interrupt the occurrence of SWDs in the corticothalamic

system, thus providing an alternative mechanism to regulate typical absence seizure activities. To our

knowledge, however, so far the precise roles of these direct basal ganglia-thalamic pathways in controlling

absence seizures are not completely established.

To address this question, we develop a realistic mean-field model for the basal ganglia-corticothalamic

(BGCT) network in the present study. Using various dynamic analysis techniques, we show that the

2

Figure 1: Framework of the basal ganglia-corticothalamic network. Neural populations include

e = excitatory pyramidal neurons; i = inhibitory interneurons; r = thalamic reticular nucleus (TRN);

s = specific relay nuclei (SRN); d1 = striatal D1 neurons; d2 = striatal D2 neurons; p1 = substantia nigra

pars reticulata (SNr); p2 = globus pallidus external (GPe) segment; ζ = subthalamic nucleus (STN).

Note that we do not model the globus pallidus internal (GPi) segment independently but consider SNr

and GPi as a signal structure in this work. Red lines with arrow heads denote the excitatory projections

mediated by glutamate receptors. Blue solid and dashed lines with round heads represent the inhibitory

projections mediated by GABAA and GABAB receptors, respectively. Compared with the traditional

model of corticothalamic system on absence seizures, the basal ganglia are also included in our biophysical

model.

absence seizures are controlled and modulated either by the isolated SNr-TRN pathway or the isolated

SNr-SRN pathway. Under suitable conditions, these two types of modulations are observed to coexist in

the same network. Importantly, in this coexist region, both low and high activation levels of SNr neurons

can suppress the occurrence of SWDs due to the competition between these two direct inhibitory basal

ganglia-thalamic pathways. These findings clearly outline a bidirectional control of absence seizures by the

basal ganglia, which is a novel phenomenon that has never been identified both in previous experimental

and modelling studies. Our results, on the one hand, further improve the understanding of the significant

role of basal ganglia in controlling absence seizure activities, and on the other hand, provide testable

hypotheses for future experimental studies.

Materials and Methods

Model

We build a biophysically based model that describes the population dynamics of the BGCT net-

work to investigate the possible roles of basal ganglia in the control of absence seizures. The network

framework of this model is inspired by recent modelling studies on Parkinson’s disease [30, 31], which is

shown schematically in Fig. 1. The network totally includes nine neural populations, which are indicated

as follows: e = excitatory pyramidal neurons; i = inhibitory interneurons; r = TRN; s = SRN; d1 =

striatal D1 neurons; d2 = striatal D2 neurons; p1 = SNr; p2 = globus pallidus external (GPe) segment;

ζ = subthalamic nucleus (STN). Similar to other modelling studies [30–32], we do not model the globus

3

pallidus internal (GPi) segment independently but consider SNr and GPi as a signal structure in the

present study, because they are reported to have closely related inputs and outputs, as well as similarities

in cytology and function. Three types of neural projections are contained in the BGCT network. For

sake of clarity, we employ different line types and heads to distinguish them (see Fig. 1). The red lines

with arrow heads denote the excitatory projections mediated by glutamate, whereas the blue solid and

dashed lines with round heads represent the inhibitory projections mediated by GABAA and GABAB,

respectively. It should be noted that in the present study the connections among different neural popula-

tions are mainly inspired by previous modelling studies [30, 31]. Additionally, we also add the connection

sending from SNr to TRN in our model, because recent anatomical findings have provided evidence that

the SNr also contains GABAergic neurons directly projecting to the TRN [27–29].

The dynamics of neural populations are characterized by the mean-field model [9, 9, 11, 33–35], which

was proposed to study the macroscopic dynamics of neural populations in a simple yet efficient way.

The first component of the mean-field model describes the average response of populations of neurons to

changes in cell body potential. For each neural population, the relationship between the mean firing rate

Qa and its corresponding mean membrane potential Va satisfies an increasing sigmoid function, given by

Qa(r, t) ≡ F [Va(r, t)] =Qmax

a

1 + exp[− π√3

(Va(r,t)−θa)σ ]

, (1)

where a ∈ A = {e, i, r, s, d1, d2, p1, p2, ζ} indicate different neural populations, Qmaxa denotes the maxi-

mum firing rate, r represents the spatial position, θa is the mean firing threshold, and σ is the threshold

variability of firing rate. If Va exceeds the threshold θa, the neural population fires action potentials with

an average firing rate Qa. It should be noted that the sigmoid shape of Qa is physiologically crucial

for this model, ensuring that the average firing rate cannot exceed the maximum firing rate Qmaxa . The

changes of the average membrane potential Va at the position r, under incoming postsynaptic potentials

from other neurons, are modeled as [9, 33–36]

DαβVa(r, t) =∑

b∈A

vab · φb(r, t), (2)

Dαβ =1

αβ[∂2

∂t2+ (α+ β)

∂

∂t+ αβ], (3)

where Dαβ is a differential operator representing the dendritic filtering of incoming signals. α and β

are the decay and rise times of cell-body response to incoming signals, respectively. vab is the coupling

strength between neural populations of type a and type b. φb(r, t) is the incoming pulse rate from the

neural population of type b to type a. For simplicity, we do not consider the transmission delay among

most neural populations in the present work. However, since the GABAB functions via second messenger

processes, a delay parameter τ is introduced to its incoming pulse rate (i.e., φb(r, t−τ)) to mimic its slow

synaptic kinetics. This results in a delay differential equation in the final mathematical description of

the BGCT model. Note that the similar modelling method has also been used in several previous studies

[13, 37].

In our system, each neural population gives rise to a field φa of pulses, which travels to other neural

population at a mean conduction velocity va. In the continuum limit, this type of propagation can be

well-approximated by a damped wave equation [9, 33–35, 38]:

1

γ2a

[∂2

∂t2+ 2γa

∂

∂t+ γ2

a − v2a∇2]φa(r, t) = Qa(r, t). (4)

Here ∇2 is the Laplacian operator (the second spatial derivative), ra is the characteristic range of axons

of type a, and γa = va/ra governs the temporal damping rate of pulses. In our model, only the axons of

4

cortical excitatory pyramidal neurons are assumed to be sufficiently long to yield significant propagation

effect. For other neural populations, their axons are too short to support wave propagation on the relevant

scales. This gives φc = F (Vc) (c = i, r, s, d1, d2, p1, p2, ζ). Moreover, as one of typical generalized seizures,

the dynamical activities of absence seizures are believed to occur simultaneously throughout the brain.

A reasonable simplification is therefore to assume that the spatial activities are uniform in our model,

which has been shown as the least stable mode in models of this class [33, 34, 36]. To this end, we ignore

the spatial derivative and set ∇2 = 0 in Eq. (4). Accordingly, the propagation effect of cortical excitatory

axonal field φe is finally given by [33, 34, 36]:

1

γ2e

[d2

dt2+ 2γe

d

dt+ γ2

e ]φe(t) = Qe(t), (5)

where γe = ve/re. For the population of cortical inhibitory interneurons, the BGCT model can be further

reduced by using Vi = Ve and Qi = Qe, which is based on the assumption that intracortical connectivities

are proportional to the numbers of synapses involved [9, 13, 33–36]. It has been demonstrated that by

making these above reductions, the developed BGCT model becomes computationally more tractable

without significant deteriorating the precision of numerical results.

We then rewrite above equations in the first-order form for all neural populations. Following above

assumptions, we use Eqs. (1)-(3) and (5) for modelling the dynamics of excitatory pyramidal neurons, and

Eqs. (1)-(3) for modelling the dynamics of other neural populations. This yields the final mathematical

description of the BGCT model given as follows:

dφe(t)

dt= φe(t), (6)

dφe(t)

dt= γ2

e [−φe(t) + F (Ve(t))]− 2γeφe(t), (7)

dX(t)

dt= X(t), (8)

dX(t)

dt= CY (t)−W (t), (9)

where

X(t) =[

Ve(t), Vd1(t), Vd2

(t), Vp1(t), Vp2

(t), Vζ(t), Vr(t), Vs(t)]T

, (10)

C = αβ, (11)

W (t) = (α+ β)X(t). (12)

In Eq. (10), the superscript T denotes transposition. The detailed expression of Y (t) for different neural

populations is represented by C1, Y1(t) and Y2(t), given by

Y (t) = C1Y1(t)− Y2(t), (13)

with

C1 =

vee vei 0 0 0 0 0 (0, 0) ves

vd1e 0 vd1d10 0 0 0 (0, 0) vd1s

vd2e 0 0 vd2d20 0 0 (0, 0) vd2s

0 0 vp1d10 0 vp1p2

vp1ζ (0, 0) 0

0 0 0 vp2d20 vp2p2

vp2ζ (0, 0) 0

vζe 0 0 0 0 vζp20 (0, 0) 0

vre 0 0 0 vrp10 0 (0, 0) vrs

vse 0 0 0 vsp10 0 (vAsr, v

Bsr) 0

, (14)

5

Y1(t) =[

φe, F (Ve), F (Vd1), F (Vd2

), F (Vp1), F (Vp2

), F (Vζ),(

F (Vr), F (Vr(t− τ)))

, F (Vs)]T

, (15)

Y2(t) =[

Ve(t), Vd1(t), Vd2

(t), Vp1(t), Vp2

(t), Vζ(t), Vr(t), Vs(t)− φn

]T

. (16)

Here the variable τ in Eq. (15) denotes the GABAB delay and the parameter φn in Eq. (16) represents

the constant nonspecific subthalamic input onto SRN.

The parameters used in our BGCT model are compatible with physiological experiments and their

values are adapted from previous studies [9, 11, 13, 30, 31, 36]. Unless otherwise noted, we use the

default parameter values listed in Table 1 for numerical simulations. Most of the default values of these

parameters given in Table 1 are based on either their nominal values or parameter ranges reported in above

literature. A small number of parameters associated with the basal ganglia (i.e., vd1d1, vp2p2

and vp2ζ)

are adjusted slightly, but still within their normal physiological ranges, to ensure our developed model

can generate the stable 2-4 Hz SWDs under certain conditions. Note that due to lack of quantitative

data, the coupling strength of the SNr-TRN pathway needs to be estimated. Considering that the SNr

sends GABAergic projections both to SRN and TRN and also both of these two nuclei are involved in

thalamus, it is reasonable to infer that the coupling strengths of these two pathways are comparable.

For simplicity, here we chose vrp1= vsp1

= −0.035 mV s by default. In the following studies, we also

change (decrease or increase) the value of vrp1several folds by employing a scale factor K (see below) to

examine how the inhibition from the SNr-TRN pathway regulates absence seizures. Additionally, during

this study, several other critical parameters (i.e., vsr , τ and vp1ζ) are also varied within certain ranges

to obtain different dynamical states and investigate their possible effects on the modulation of absence

seizures.

Data analysis

In the present study, several data analysis methods are employed to quantitatively evaluate the dy-

namical states as well as the properties of SWDs generated by the model. To reveal critical transitions

between different dynamical states, we perform the bifurcation analysis for several key parameters of the

model. For one specific parameter, the bifurcation diagram is simply obtained by plotting the “stable”

local minimum and maximum values of cortical excitatory axonal fields (i.e., φe) over changes in this

parameter [11, 39]. To this end, all simulations are executed for sufficiently long time (10 seconds of

simulation time, after the system reaches its stable time series), and only the local minimum and maxi-

mum values obtained from the latter stable time series are used. Using the above bifurcation analysis, we

can also easily distinguish different dynamical states for combined parameters. Such analysis technique

allows us to further identify different dynamical state regions in the two-parameter space (for example,

see Fig. 2D). On the other hand, the power spectral analysis is used to estimate the dominant frequency

of neural oscillations. To do this, the power spectral density is obtained from the time series φe (over a

period of 10 seconds) by using the fast Fourier transform. Then, the maximum peak frequency is defined

as the dominant frequency of neural oscillations. It should be noted that, by combining the results of

both the state and frequency analysis, we can outline the SWD oscillation region that falls into the 2-4

Hz frequency range in the two-parameter space (for example, see the asterisk region in Fig. 2E). More-

over, we calculate the mean firing rates (MFRs) for several key neural populations in some figures. To

compute the MFRs, all corresponding simulations are performed up to 25 seconds and the data from 5

to 25 seconds are used for statistical analysis. To obtain convincing results, we carry out 20 independent

simulations with different random seeds for each experimental setting, and report the averaged result as

the final result. Finally, in some cases, we also compute the low and high triggering mean firing rates

(TMFRs) for SNr neurons. In the following simulations, we find that the mean firing rate of SNr neurons

6

A: Maximum firing rate

Symbol Description Value References

Qmaxe , Qmax

i Cortical maximum firing rate 250 Hz [9, 11, 13]

Qmaxd1

, Qmaxd2

Striatum maximum firing rate 65 Hz [30, 31]

Qmaxp1

SNr maximum firing rate 250 Hz [30, 31]

Qmaxp2

GPe maximum firing rate 300 Hz [30, 31]

Qmaxζ STN maximum firing rate 500 Hz [30, 31]

Qmaxs SRN maximum firing rate 250 Hz [9, 11, 13]

Qmaxr TRN maximum firing rate 250 Hz [9, 11, 13]

B: Mean firing threshold

Symbol Description Value References

θe, θi Mean firing threshold of cortical populations 15 mV [9, 11, 13, 36]

θd1 , θd2 Mean firing threshold of striatum 19 mV [30, 31]

θp1 Mean firing threshold of SNr 10 mV [30, 31]

θp2 Mean firing threshold of GPe 9 mV [30, 31]

θζ Mean firing threshold of STN 10 mV [30, 31]

θs Mean firing threshold of SRN 15 mV [9, 11, 13, 36]

θr Mean firing threshold of TRN 15 mV [9, 11, 13, 36]

C: Coupling strength

Symbol Source Target Value References

vee Excitatory pyramidal neurons Excitatory pyramidal neurons 1 mV s [11, 13]

−vei Inhibitory interneurons Excitatory pyramidal neurons 1.8 mV s [11, 13]

vre Excitatory pyramidal neurons TRN 0.05 mV s [9, 13]

vrs SRN TRN 0.5 mV s [9, 13]

−vA,Bsr TRN SRN 0.4− 2 mV s [9, 36]

vd1e Excitatory pyramidal neurons Striatal D1 neurons 1 mV s [30, 31]

−vd1d1 Striatal D1 neurons Striatal D1 neurons 0.2 mV s [30, 31]

vd1s SRN Striatal D1 neurons 0.1 mV s [30, 31]

vd2e Excitatory pyramidal neurons Striatal D2 neurons 0.7 mV s [30, 31]

−vd2d2 Striatal D2 neurons Striatal D2 neurons 0.3 mV s [30, 31]

vd2s SRN Striatal D2 neurons 0.05 mV s [30, 31]

−vp1d1 Striatal D1 neurons SNr 0.1 mV s [30, 31]

−vp1p2 GPe SNr 0.03 mV s [30, 31]

vp1ζ STN SNr 0− 0.6 mV s [30, 31]

−vp2d2 Striatal D2 neurons GPe 0.3 mV s [30, 31]

−vp2p2 GPe GPe 0.075 mV s [30, 31]

vp2ζ STN GPe 0.45 mV s [30, 31]

−vζp2 GPe STN 0.04 mV s [30, 31]

ves SRN Excitatory pyramidal neurons 1.8 mV s [9, 13]

vse Excitatory pyramidal neurons SRN 2.2 mV s [9]

vζe Excitatory pyramidal neurons STN 0.1 mV s [30, 31]

−vsp1 SNr SRN 0.035 mV s [30, 31]

−vrp1 SNr TRN 0.035 mV s Estimated

D: Other parameters

Symbol Description Value References

γe Cortical damping rate 100 Hz [9, 11, 13]

τ Time delay due to slow synaptic kinetics of GABAB 50 ms [13]

α Synaptodendritic decay time constant 50 s−1 [9, 11, 13, 36]

β Synaptodendritic rise time constant 200 s−1 [9, 11, 13, 36]

σ Threshold variability of firing rate 6 mV [9, 11, 36]

φn Nonspecific subthalamic input onto SRN 2 mV s [9, 11, 36]

Table 1: Model parameters employed in the present study. Unless otherwise noted, we use these default

parameter values for simulations.

7

is increased with the growth of the excitatory coupling strength vp1ζ , which serves as a control parameter

to modulate the activation level of SNr in our work (see the Results section). Based on this property,

the low and high TMFRs can be determined by the mean firing rates of SNr neurons occurring at the

boundaries of the typical region of 2-4 Hz SWDs (for example, see the black dashed lines in Fig. 3B).

Numerical simulation

All network simulations are written and performed under the MATLAB environment. The aforemen-

tioned dynamical equations are integrated by using the standard fourth-order Runge-Kutta method, with

a fixed temporal resolution of h = 0.05 ms [40]. In additional simulations, it turns out that the chosen

integration step is sufficiently small to ensure the numerical accuracy of our developed BGCT model. The

basic computer code used in the present study is provided as supplementary information to this paper

(we also provide a XPPAUT code for comparison [41]; see Text S1 and S2).

Results

Slow kinetics of GABAB receptors in TRN create absence seizure activities

for the BGCT model

Previous studies have suggested that the slow kinetics of GABAB receptors in TRN are a candidate

pathological factor contributing to the generation of absence seizures both in animal experiments and

biophysical models of corticothalamic network [7, 13, 42, 43]. To explore whether this mechanism also

applies to the developed BGCT model, we perform one-dimensional bifurcation analysis for the inhibitory

coupling strength −vsr and the delay parameter τ , respectively. The corresponding bifurcation diagrams

and typical time series of φe are depicted in Figs. 2A-2C, which reveal that different dynamical sates

emerge in our system for different values of −vsr and τ . When the coupling strength −vsr is too weak,

the inhibition from TRN cannot effectively suppress the firing of SRN. In this case, due to the strong

excitation from pyramidal neurons, the firing of SRN rapidly reaches a high level after the beginning

of the simulation. Such high activation level of SRN in turn drives the firing of cortical neurons to

their saturation states within one or two oscillation periods (region I). As the coupling strength −vsr

grows, the inhibition from TRN starts to affect the firing of SRN. For sufficiently long τ , this causes

our model to successively undergo two different oscillation patterns. The first one is the SWD oscillation

pattern, in which multiple pairs of maximum and minimum values are found within each periodic complex

(region II). Note that this oscillation pattern has been extensively observed on the EEG recordings of

real patients during absence seizures [1]. The other one is the simple oscillation pattern, in which only

one pair of maximum and minimum values appears within each periodic complex (region III). However,

if the coupling strength −vsr is too strong, the firing of SRN is almost completely inhibited by TRN. In

this situation, the model is kicked into the low firing region and no oscillation behavior can be observed

anymore (region IV). Additionally, we also find that the model dynamics are significantly influenced by

the GABAB delay, and only sufficiently long τ can ensure the generation of SWDs in the developed model

(see Fig. 2B).

To check whether our results can be generalized within a certain range of parameters, we further carry

out the two-dimensional state analysis in the (−vsr, τ) panel. As shown in Fig. 2D, the whole (−vsr, τ)

panel is divided into four state regions, corresponding to those regions identified above. Unsurprisingly, we

find that the BGCT model can generate the SWD oscillation pattern only for appropriately intermediate

−vsr and sufficiently long τ . This observation is in consistent with our above finding, demonstrating

8

0.5 1 1.5

101

102

−vsr

(mV s)

φ e (H

z)

20 40 60

10

20

30

40

50

Dealy τ (ms)

φ e (H

z)

0.40 1.08 1.7620

45

70

τ (m

s)

State region

−vsr

(mV s)

0.40 1.08 1.7620

45

70

τ (m

s)

Dominant frequency (Hz)

−vsr

(mV s)

0

250φ e (H

z) 0

40φ e (H

z)

0 1 2

0

20φ e (H

z)

Time (s)0 1 2

0

15

Time (s)

φ e (H

z)

0

2

4

6

8

B

I

III

IV

D E

C

II

III

IV

I

III

II

τ = 50 ms

Saturation (I)

Simple oscillation (III)

SWD oscillation (II)

Low firing (IV)

II

A

vsr

= −1.06 mV s

Figure 2: Absence seizure activities induced by the slow kinetics of GABAB receptors in

TRN. A, B: Bifurcation diagrams of φe as a function of the TRN-SRN inhibitory coupling strength

−vsr (A) and the delay parameter τ (B), respectively. Four different dynamical states can be observed

from the time series of φe, which are: the saturation state (I), the SWD oscillation state (II), the simple

oscillation state (III) and the low firing state (IV). C: Typical time series of φe correspond to the above

four dynamical states. Here we set τ = 50 ms and chose vsr = −0.48 mV s (I), vsr = −1 mV s (II),

vsr = −1.48 mV s (III), vsr = −1.6 mV s (IV), respectively. The colors in bifurcation diagrams (A)

and (B) correspond to the typical time series plotted in (C). D, E: The state analysis (D) and frequency

analysis (E) in the (−vsr, τ) panel. Different colors in (D) represent different dynamical state regions,

corresponding to those dynamical states given in (A), (B) and (C). The asterisk (“∗”) regions surrounded

by black dashed lines in (D) and (E) represent the SWD oscillation regions falling into the 2-4 Hz frequency

range. The other symbols in (D) and (E) are linked to parameter values used for different typical time

series in (C): I (“•”), II (“�”), III (“H”), and IV (“⋆”). For all simulations, we set vp1ζ = 0.3 mV s.

9

the generalizability of our above results. To estimate the frequency characteristics of different oscillation

patterns, we compute the dominant frequency based on the spectral analysis in the (−vsr, τ) panel. For

both the simple and SWD oscillation patterns, the dominant frequency is influenced by τ and −vsr, and

increasing their values can both reduce the dominant frequency of neural oscillations (Fig. 2E). However,

compared to −vsr, our results indicate that the GABAB delay may have a more significant effect on the

dominant oscillation frequency (Fig. 2E). By combining the results in Figs. 2D and 2E, we roughly outline

the SWD oscillation region that falls into the 2-4 Hz frequency range (asterisk region). It is found that

most of, but not all, the SWD oscillation region is contained in this specific region. Here we emphasize

the importance of this specific region, because the SWDs within this typical frequency range is commonly

observed during the paroxysm of absence epilepsy in human patients [1, 2].

Why can the slow kinetics of GABAB receptors in TRN induce absence seizure activities? Anatomi-

cally, the SRN neurons receive the TRN signals from the inhibitory pathway mediated by both GABAA

and GABAB receptors. Under suitable condition, the double suppression caused by these two types

of GABA receptors occurring at different time instants may provide an effective mechanism to create

multiple firing peaks for the SRN neurons (see below). Such firing pattern of SRN in turn impacts the

dynamics of cortical neurons, thus leading to the generation of SWDs. It should be noted that, during the

above processes, both τ and −vsr play critical roles. In each oscillation period, after the GABAA-induced

inhibition starts to suppress the firing of SRN neurons, these neurons need a certain recovery time to

restore their mean firing rate to the rising state. Theoretically, if this recovery time is shorter than the

GABAB delay, another firing peak can be introduced to SRN neurons due to the latter GABAB-induced

inhibition. The above analysis implies that our model requires a sufficient long GABAB delay to ensure

the occurrence of SWDs. However, as described above, too long τ is also a potential factor which may

push the dominant frequency of SWDs beyond the typical frequency range. For a stronger −vsr, the

inhibition caused by GABAA is also strong. In this situation, it is obvious that the SRN neurons need a

longer time to restore their firing rate. As a consequent, a relatively longer τ is required for the BGCT

model to ensure the occurrence of SWDs for stronger −vsr (see Fig. 2D).

These findings provide consistent evidence that our developed BGCT model can replicate the typical

absence seizure activities utilizing previously verified pathological mechanism. Because we do not change

the normal parameter values for basal ganglia during above studies, our results may also indicate that,

even though the basal ganglia operate in the normal state, the abnormal alteration within the corticotha-

lamic system may also trigger the onset of absence epilepsy. Throughout the following studies, we set

τ = 50 ms for all simulations. For this choice, the delay parameter τ is within the physiological range

and modest, allowing the generation of SWD oscillation pattern while preserving its dominant frequency

around 3 Hz in most considered parameter regions. It should be noted that, in additional simulations, we

have shown that by slightly tuning the values of several parameters our developed BGCT model is also

powerful to reproduce many other typical patterns of time series, such as the alpha and beta rhythms

(see Figure S1), which to a certain extent can be comparable with real physiological EEG signals [9, 36].

Control of absence seizures by the isolated SNr-TRN pathway

Using the developed BGCT model, we now investigate the possible roles of basal ganglia in controlling

absence seizure activities. Here we mainly concentrate on how the activation level of SNr influence the

dynamics generated by the model. This is because, on the one hand, the SNr is one of chief output nucleus

of the basal ganglia to thalamus, and on the other hand, its firing activity has been found to be highly

associated with the regulation of absence seizures [21, 22]. To this end, the excitatory coupling strength

vp1ζ is employed to control the activation level of SNr and a three-step strategy is pursued in the present

10

work. In this and next subsections, we assess the individual roles of two different pathways emitted

from SNr to thalamus (i.e., the SNr-TRN and SNr-SRN pathways) in the control of absence seizures and

discuss their corresponding biophysical mechanisms, respectively. In the final two subsections, we further

analyze the combination effects of these two pathways on absence seizure control and extend our results

to more general cases.

To explore the individual role of the SNr-TRN pathway, we estimate both the state regions and

frequency characteristics in the (−vsr, vp1ζ) panel. Note that during these investigations the SNr-SRN

pathway is artificially blocked (i.e., vsp1= 0 mV s). With this “naive” method, the modulation of

absence seizure activities by the SNr-SRN pathway is removed and the effect caused by the SNr-TRN

pathway is theoretically amplified to the extreme. Similar to previous results, we find that the whole

(−vsr, vp1ζ) panel can be also divided into four different regions (Fig. 3A). These regions are the same

as those defined above. For weak inhibitory coupling strength −vsr, increasing the excitatory coupling

strength vp1ζ moves the model dynamics from the SWD oscillation state to the saturation state. Here we

have to notice that the saturation state is a non-physiological brain state even though it does not belong

to typical seizure activities. In strong −vsr region, the suppression of SWDs is observed by decreasing

the excitatory coupling strength vp1ζ , suggesting that inactivation of SNr neurons may result in seizure

termination through the SNr-TRN pathway (Fig. 3A, right side). For strong enough inhibitory coupling

strength −vsr, such suppression effect is rather remarkable that sufficiently low activation of SNr can

even kick the network dynamics into the low firing region (compare the results in Figs. 3C and 3D).

The SNr-TRN pathway induced SWD suppression is complicated and its biophysical mechanism is

presumably due to competition-induced collision. On the one side, the decrease of excitatory coupling

strength vp1ζ inactivates the SNr (Fig. 3E, top panel), which should potentially enhance the firing of TRN

neurons. On the other side, however, increasing the activation level of TRN tends to suppress the firing of

SRN, which significantly reduces the firing of cortical neurons and in turn inactivates the TRN neurons.

Furthermore, the inactivation of cortical neurons also tends to reduce the firing level of TRN neurons. As

the excitatory coupling strength vp1ζ is decreased, the collision caused by such complicated competition

and information interactions finally leads to the inactivation for all the TRN, SRN, and cortical neurons

(Fig. 3E, bottom panel), which potentially provides an effective mechanism to destabilize the original

pathological balance within the corticothalamic system, thus causing the suppression of SWDs.

Indeed, we find that not only the dynamical state but also the oscillation frequency is greatly impacted

by the activation level of SNr, through the SNr-TRN pathway. For both the simple and SWD oscillation

patterns, increasing the excitatory strength vp1ζ can enhance their dominant frequencies. The combined

results of Figs. 3A and 3B reveal that, for a fixed −vsr, whether the model can generate the SWDs within

the typical 2-4 Hz is determined by at least one and often two critical values of vp1ζ (Fig. 3B, asterisk

region). Because the activation level of SNr is increased with the growth of vp1ζ , this finding further

indicates that, due to effect of the SNr-TRN pathway, the model might exist the corresponding low and

high triggering mean firing rates (TMFRs) for SNr neurons (Fig. 3E, dashed lines). If the long-term

mean firing rate of SNr neurons falls into the region between these two TMFRs, the model can highly

generate typical 2-4 Hz SWDs as those observed on the EEG recordings of absence epileptic patients. In

Fig. 3F, we plot both the low and high TMFRs as a function of the inhibitory coupling strength −vsr.

With the increasing of −vsr, the high TMFR grows rapidly at first and then reaches a plateau region,

whereas the low TMFR almost linearly increases during this process. Consequently, it can be seen that

these two critical TMFRs approach each other as the inhibitory coupling strength −vsr is increased until

they almost reach an identical value (Fig. 3F).

The above findings indicate that the SNr-TRN pathway may play a vital role in controlling the

absence seizures and appropriately reducing the activation level of SNr neurons can suppress the typical

11

State region

−vsr

(mV s)

v p 1ζ (m

V s

)

0.48 1.24 2.00

0.3

0.6

Dominant frequency (Hz)

−vsr

(mV s)

v p 1ζ (m

V s

)

0.48 1.24 2.00

0.3

0.6

0 0.2 0.4 0.60

20

40

60

vp

1ζ (mV s)

φ e (H

z)

0 0.2 0.4 0.60

20

40

60

vp

1ζ (mV s)

φ e (H

z)

0 0.2 0.4 0.6

10

MF

R (

Hz)

vp

1ζ (mV s)

0.5 1 1.5 20

25

50

75

100

TM

FR

(H

z)

−vsr

(mV s)

HighLow

0

1

2

3

4

5

10

150

vsr

= −1.20 mV s

B

D

φp1 HighLow

A

C

E

vsr

= −1.44 mV s

III

II

III

II

IV

φe

φs

φr

Suppression

I

IIIIIIV

F

Figure 3: Control of absence seizures by the isolated SNr-TRN pathway. A, B: The state

analysis (A) and frequency analysis (B) in the (−vsr , vp1ζ) panel. Here −vsr is the inhibitory coupling

strength of the TRN-SRN pathway, whereas vp1ζ is the excitatory coupling strength of the STN-SNr

pathway. Different colors in (A) represent different dynamical state regions: the saturation region (I),

the SWD oscillation region (II), the simple oscillation region (III) and the low firing region (IV). The

suppression of SWDs appears to the right of the white dashed line in (A), where the down arrow indicates

that the SWD oscillation can be inhibited by decreasing vp1ζ . The asterisk (“∗”) region surrounded by

black dashed lines in (B) denotes the typical 2-4 Hz SWD oscillation region. C, D: Bifurcation diagrams

of φe as a function of vp1ζ for different −vsr. The strengths of the inhibitory projections from the TRN

to SRN are set as vsr = −1.20 mV s (C) and vsr = −1.44 mV s (D), respectively. Different colors in

(C) and (D) represent different dynamical state regions, corresponding to those in phase diagram (A). E:

The mean firing rates (MFRs) of several key neural populations as a function of vp1ζ , with vsr = −1.44

mV s. Here four neural populations are considered: SNr (“·”), excitatory pyramidal neurons (“∗”), SRN

(“◦”) and TRN (“�”). Two black dashed lines in (E) represent the occurring positions of the low and

high triggering mean firing rates (TMFRs), respectively. F: The low and high TMFRs as a function of

−vsr. For all simulations, the SNr-SRN pathway is artificially blocked (i.e., vsp1= 0 mV s).

12

2-4 Hz SWDs. The similar antiepileptic effect induced by inactivating the SNr has been widely reported

in previous electrophysiological experiments based on both genetic absence epilepsy rats and tottering

mice [21–24]. Note that, however, in literature such antiepileptic effect by reducing the activation of SNr

is presumed to be accomplished through the indirect SNr-TRN pathway relaying at superior colliculus

[21, 22]. Our computational results firstly suggest that such antiepileptic process can be also triggered by

the direct SNr-TRN GABAergic projections. Combining these results, we postulate that for real absence

epileptic patients both of these two pathways might work synergistically and together provide a stable

mechanism to terminate the onset of absence epilepsy.

Control of absence seizures by the isolated SNr-SRN pathway

We next turn on the SNr-SRN pathway and investigate whether this pathway is also effective in the

control of absence seizures. Similar to the previous method, we artificially block the SNr-TRN pathway

(i.e., vrp1= 0 mV s) to enlarge the effect of the SNr-SRN pathway to the extreme. Figure 4A shows the

two-dimensional state analysis in the (−vsr, vp1ζ) panel, and again the whole panel is divided into four

different state regions. Compared to the results in Fig. 3A, the suppression of SWDs appears in a relatively

weaker −vsr region by increasing the excitatory coupling strength vp1ζ . This finding suggests that the

increase in the activation of SNr can also terminate the SWDs, but through the SNr-SRN pathway.

For relatively weak −vsr within the suppression region, the SNr-SRN pathway induced suppression is

somewhat strong. In this case, the high activation level of SNr directly kicks the network dynamics

into the low firing region, without undergoing the simple oscillation state (Fig. 4C2 and compare with

Fig. 4C3). Note that this type of state transition is a novel one which has not been observed in the

SWD suppression caused by the SNr-TRN pathway. For relatively strong −vsr within the suppression

region, the double peak characteristic of SWDs generated by our model is weak. In this situation, as the

inhibitory coupling strength −vsr is increased, we observe that the network dynamics firstly transit from

the SWD oscillation state to the simple oscillation state, and then to the low firing state (Fig. 4C3).

To understand how the SNr-SRN pathway induced SWD suppression arises, we present the mean firing

rates of several key neural populations within the corticothalamic system, as shown in Fig. 4D. It can

be seen that increasing the strength vp1ζ significantly improves the activation level of SNr (Fig. 4D, top

panel), which in turn reduces the firing of SRN neurons (Fig. 4D, bottom panel). The inactivation of SRN

neurons further suppresses the mean firing rates for both cortical and TRN neurons (Fig. 4D, bottom

panel). These chain reactions lead to the overall inhibition of firing activities in the corticothalamic

system, which weakens the double peak shaping effect due to the slow kinetics of GABAB receptors in

TRN. For strong vp1ζ , such weakening effect is considerable, thus causing the suppression of SWDs. Our

results provide the computational evidence that high activation of SNr can also effectively terminate

absence seizure activities by the strong inhibition effect from the SNr-SRN pathway. Compared to the

SWD suppression induced by the SNr-TRN pathway, it is obvious that the corresponding biophysical

mechanism caused by the SNr-SRN pathway is simpler and more direct.

Moreover, our two-dimensional frequency analysis indicates that the dominant frequency of neural

oscillations depends on the excitatory coupling strength vp1ζ (see Fig. 4B). For a constant −vsr, progres-

sive increase of vp1ζ reduces the dominant frequency, but not in a very significant fashion. Thus, we find

that almost all the SWD oscillation region identified in Fig. 4A falls into the typical 2-4 Hz frequency

range (Fig. 4B, asterisk region). Unlike the corresponding results presented in previous subsection, the

combination results of Figs. 4A and 4B demonstrate that the BGCT model modulated by the isolated

SNr-SRN pathway only exhibits one TMFR for SNr neurons. For a suitably fixed strength −vsr, the

generation of SWDs can be highly triggered when the mean firing rate of SNr neurons is lower than this

13

State regionv p 1ζ (

mV

s)

−vsr

(mV s)0.4 0.84 1.28

0

0.6

1.2

Dominant frequency (Hz)

−vsr

(mV s)

v p 1ζ (m

V s

)

0.4 0.84 1.280

0.6

1.2

0

1

v p 1ζ (m

V s

)

0.5 0.6 0.7 0.8 0.90

50

100

150

200

TM

FR

(H

z)

−vsr

(mV s)

0

1

2

3

4

0

45φ e (H

z)

0 2 4 6 8

0

25φ e (H

z)

time (s)

0 0.5 10

15

30

MF

R (

Hz)

vp

1ζ (mV s)

0

250

A B

IIIIV

I

C

C1

C2

C3

E

II

vsr

= −0.72 mV s

vsr

= −0.92 mV s

φr

φs

φp1

φe

DTMFR

Suppression

Figure 4: Control of absence seizures by the isolated SNr-SRN pathway. A, B: The state analysis

(A) and frequency analysis (B) in the (−vsr, vp1ζ) panel. Here −vsr is the inhibitory coupling strength

of the TRN-SRN pathway, whereas vp1ζ is the excitatory coupling strength of the STN-SNr pathway.

Similar to previous results, four different dynamical state regions are observed: the saturation region (I),

the SWD oscillation region (II), the simple oscillation region (III) and the low firing region (IV), which

correspond to those defined in Fig.2 (D). The region between two dashed lines in (A) represents the

suppression region of SWDs, where the up arrow indicates that the SWD oscillation can be inhibited by

increasing vp1ζ . The asterisk (“∗”) region surrounded by dashed lines in (B) denotes the typical 2-4 Hz

SWD oscillation region. C: Two different types of SWD suppressions caused by linearly increasing vp1ζ .

Top (C1): The value of vp1ζ as a function of time. Middle (C2): Corresponding φe trace for vsr = −0.72

mV s. Bottom (C3): Corresponding φe trace for vsr = −0.92 mV s. D: The MFRs of several key neural

populations as a function of vp1ζ , with vsr = −0.72 mV s. Here four neural populations are considered:

SNr (“·”), excitatory pyramidal neurons (“∗”), SRN (“◦”) and TRN (“�”). The black dashed line in

(D) represents the occurring position of TMFR. E: The TMFR as a function of −vsr. Note that the

SNr-TRN pathway is artificially blocked (i.e., vrp1= 0 mV s) for all simulations.

14

critical firing rate (Fig. 4D, dashed line). With the increasing of −vsr, we observe that this TMFR rapidly

reduces from a high value to a low value (Fig. 4E). Note that this decreasing tendency is in contrast with

our previous finding based on the model modulated by the isolated SNr-TRN pathway (compared to the

results in Fig. 3F).

Taken together, these observations suggest that increasing the activation level of SNr neurons also

significantly influences the dynamics of the corticothalamic system and causes the suppression of absence

seizure activities. To the best of our knowledge, this is a new finding that underscores the importance

of the direct inhibitory SNr-SRN pathway in controlling and modulating absence seizure activities. It is

reasonable to believe that several other external factors, which are able to enhance the activation level

of SNr, may also result in the termination of absence seizures due to the similar mechanism.

Competition-induced bidirectional control of absence seizures by the basal

ganglia

So far, we have confirmed that the absence seizure activities generated by the BGCT model can be

inhibited by either the isolated SNr-TRN pathway or the isolated SNr-SRN pathway, through different

biophysical mechanisms. In real brain, however, both of these pathways should be available and work

together at the same time. Thus, an important and naturally arising question is whether these two types

of seizure control can coexist in the same network, and if possible, whether this feature can be maintained

in certain range of parameters. To address this issue, we introduce a scale factor K and set the coupling

strength vrp1= Kvsp1

. By utilizing this method, we can flexibly control the relative coupling strength

between the SNr-TRN and SNr-SRN pathways and discuss the combination roles of these two inhibitory

pathways in detail.

Figure 5A shows the state analysis in the (K, vp1ζ) panel with vsr = −1.08 mV s. Unlike the

previous results, here we only discover three dynamical state regions, which correspond to: the SWD

oscillation state (II), the simple oscillation state (III), and the low firing state (IV). The disappearance

of the saturation state is at least due to the following two reasons: (1) the double suppression from

the SNr-TRN and SNr-SRN pathways and (2) the relatively strong inhibitory effect from TRN to SRN.

As shown in Fig. 5A, the phenomenon of the SWD suppression appears in the strong K region. For

relatively weak K within this suppression region, both increasing and decreasing the activation of SNr

neurons from the normal level effectively inhibit the generation of SWDs (see Fig. 5A). Such bidirectional

suppression behavior can be attributed to the effective competition between the SNr-TRN and SNr-SRN

pathways. As the scale factor K is increased, the enhancement of the SNr-TRN pathway breaks the

original competition balance between these two inhibitory pathways. During this process, the inhibition

from the SNr-TRN pathway progressively dominates the model dynamics. Accordingly, for sufficiently

strong K, the suppression of SWDs is only found by lowing the activation of SNr (Fig. 5A), which is

consistent with our previous critical observation given in Fig. 3A.

However, as described above, the increase in the excitatory coupling strength vp1ζ significantly reduces

the dominant frequency of SWDs, through the SNr-TRN pathway. Therefore, although enhancing the

activation level of SNr neurons cannot inhibit the SWDs directly, it tends to push the dominant frequency

of SWDs below 2 Hz (Fig. 5B). By combining the results of Figs. 5A and 5B, we successfully outline the

SWD oscillation region that falls into the 2-4 Hz frequency range (asterisk region). In this typical SWD

region, the model exhibits both the low and high TMFRs of the SNr neurons for a fixed K (Figs. 5B

and 5C). The generation of SWDs within the typical frequency range can be highly triggered if the

mean firing rate of SNr neurons is between these two critical TMFRs (Fig. 5C). Nevertheless, due to

the competition between the SNr-TRN and SNr-SRN pathways, the firing activities of the TRN, SRN,

15

State region

Scale factor K

v p 1ζ (m

V s

)

0 0.5 10.03

0.3

3.0

0.4 0.6 0.80

50

100

150

200

250

TM

FR

(H

z)

Scale factor K

Dominant frequency (Hz)

Scale factor K

v p 1ζ (m

V s

)

0 0.5 10.03

0.3

3.0

0

1

2

3

0.03 0.3 30

10

20

vp

1ζ (mV s)

MF

R (

Hz)

0

250

HighLow

A B

C

φr

φe

φs

IV

II

Suppression

D

III

φp

1High

Low

Figure 5: Control of absence seizures by the combination effects of the SNr-TRN and SNr-

SRN pathways. A, B: The state analysis (A) and frequency analysis (B) in the (K, vp1ζ) panel. Here

K is the scale factor, and vp1ζ is the excitatory coupling strength of the STN-SNr pathway. Unlike

previous results, only three dynamical state regions are observed in the phase diagram (A): the SWD

oscillation region (II), the simple oscillation region (III) and the low firing region (IV), corresponding

to the dynamical states defined in Fig. 2(D). For relatively weaker scale factor K, both increase and

decrease in the activation level of SNr can inhibit the SWDs (double arrow, bidirectional suppression);

whereas for sufficiently strong K, only reducing the activation level of SNr suppresses the SWDs (single

arrow, unidirectional suppression ). In (A), the white dashed line represents the boundary of suppression

region, and the red dashed line stands for the demarcation between the bidirectional and unidirectional

suppression regions. The asterisk (“∗”) region surrounded by dashed lines in (B) denotes the SWD oscil-

lation region that falls into the 2-4 Hz frequency range. C: The MFRs of several key neural populations

as a function of vp1ζ , with the scale factor K = 0.6. Here four neural populations are considered: SNr

(“·”), excitatory pyramidal neurons (“∗”), SRN (“◦”) and TRN (“�”). Two black dashed lines in (C)

represent the occurring positions of low and high triggering mean firing rates (TMFRs), respectively. D:

The low and high TMFRs as a function of K. For all simulations, the coupling strength of the TRN-SRN

pathway is set as vsr = −1.08 mV s.

16

vsr

= −1 mV s

v p 1ζ (m

V s

)

0 0.5 10.03

0.3

3

vsr

= −1.04 mV s

0 0.5 10.03

0.3

3

vsr

= −1.12 mV s

0 0.5 10.03

0.3

3

vsr

= −1.24 mV s

0.5 10.03

0.3

3

Scale factor K

v p 1ζ (m

V s

)

0 0.5 10.03

0.3

3

Scale factor K0 0.5 1

0.03

0.3

3

Scale factor K0 0.5 1

0.03

0.3

3

Scale factor K0.5 1

0.03

0.3

3

(Hz)0

1

2

3

4

IIIII

Increase

B

A

A2

A3 A

4

B3 B

4B

2B

1

A1

IV

Figure 6: Effect of inhibitory coupling strength −vsr on the control of absence seizures by

the SNr-TRN and SNr-SRN pathways. Tow-dimensional state analysis (A) and corresponding

frequency analysis (B) in the (K, vp1ζ) panel for different values of vsr . Here only three dynamical state

regions are observed in (A): the SWD oscillation region (II), the simple oscillation region (III) and the low

firing region (IV). In (A1)-(A4), the regions marked by diamonds denote the whole suppression regions of

SWDs, the white dashed lines represent the boundaries of suppression regions, and the red dashed lines

stand for the demarcations between the bidirectional (double arrow) and unidirectional (single arrow)

suppression regions. In (B1)-(B4), the asterisk (“∗”) regions surrounded by dashed lines denote the

typical 2-4 Hz SWD oscillation regions. From left to right, the strengths of inhibitory projections from

the TRN to SRN are: vsr = −1 mV s (A1, B1), vsr = −1.04 mV s (A2, B2), vsr = −1.12 mV s (A3,

B3), and vsr = −1.24 mV s (A4, B4), respectively. For better showing and comparing the bidirectional

suppression regions among different subfigures, here we mainly consider the scale factor for all values of

vsr within the same and small interval from 0 to 1. As an additional comparison, both the state analysis

and frequency analysis for vsr = −1.20 mV s in a relatively larger K interval is given in Figure S2.

17

and cortical neurons become more complicated, compared to the cases induced by any isolated pathway.

For a constant K, such competition creates a bell-shaped MFR curve for each key neural population by

tuning the excitatory coupling strength vp1ζ , (see Fig. 5C). To further investigate how the scale factor

impacts the low and high TMFRs, we plot these two TMFRs as a function of K in Fig. 5D. With the

increasing of relative strength K, both of these two critical TMFRs are rapidly changed but in different

fashions. The high TMFR is increased from a relatively low value to saturation, whereas the low TMFR

is reduced from a relatively high value to 0 (see Fig. 5D). Obviously, such opposite tendencies of these

two TMFRs are attributed to the combination effects of the SNr-TRN and SNr-SRN pathways.

Furthermore, we also find that both the suppression of SWDs and the typical 2-4 Hz SWD region are

shaped by the strength of inhibitory projections from the TRN to SRN. In Figs. 6A and 6B, we perform

a series of two-dimensional state and frequency analysis in the (K, vp1ζ) panel for different values of −vsr.

When the inhibitory coupling strength −vsr is too weak, the SWDs generated by the corticothalamic

system is mainly controlled by the SNr-SRN pathway. In this case, the suppression of SWDs is observed

in the intermediate K region and only increasing the activation level of SNr can effectively inhibit the

SWDs (Fig. 6A). As the coupling strength −vsr is increased, the inhibition from SNr-TRN pathway starts

to influence the model dynamics. This introduces the competition between the SNr-SRN and SNr-TRN

pathways, leading to the emergence of the bidirectional suppression of SWDs for intermediateK (Fig. 6A).

It is obvious that the higher the coupling strength −vsr, the stronger the inhibition effect caused by the

SNr-TRN pathway. With increasing the value of −vsr, such strengthened inhibition significantly moves

the boundary of the low TMFR toward higher values of vp1ζ , and thus notably shrinks the region of

SWDs within the typical frequency range of 2-4 Hz (Fig. 6B).

These above observations emphasize the importance of the combination role of both the SNr-TRN and

SNr-SRN pathways on the control of absence seizure activities. Quite remarkably, we observe that the

bidirectional suppression of SWDs emerges under suitable conditions, that is, both increasing and reducing

the activation levels of SNr neurons effectively suppress the SWDs. Such bidirectional suppression is

determined and modulated by both the relative strength of these two inhibitory pathways and the strength

of inhibitory projections from the TRN to SRN. This novel finding indicates the possible bidirectional

control of absence seizures by the basal ganglia, which is induced by the competition between the SNr-

TRN and SNr-SRN pathways. In additional simulations, we have found that several other factors, which

can effectively change the activation level of SNr, can also lead to the bidirectional suppression of SWDs

due to the similar mechanism, further demonstrating the generality of our results.

Bidirectional control of absence seizures by the basal ganglia might be ex-

tended to other pathological factors

In above subsections, we focused on one specific pathological factor and have computationally shown

that the basal ganglia may bidirectionally control and modulate the typical absence seizure activities (i.e.,

the SWDs) induced by the slow synaptic kinetics of GABAB in TRN. A natural question to ask is whether

such bidirectional control feature caused by the basal ganglia is a generalized regulatory mechanism for

absence seizures. We argue that this might be true. To check this postulation, at least one additional

SWD generation mechanism should be introduced into our model, and we need to examine whether the

bidirectional control of absence seizures by the basal ganglia is also available for the new pathological

factor. In literature, there are several other theories that are associated with the generation mechanisms

of SWDs. A boldly accepted one is related to the transmission delay between the cerebral cortex and

thalamus and, specifically, it has been found that suitably choosing such transmission delay can drive

the corticothalamic system produce the SWDs [11, 33]. To apply this pathological factor in our model,

18

State region

v p 1ζ (m

V s

)

0.5 1 1.5 2 2.50.095

0.48

3.0

Dominat frequency (Hz)

Scale factor K

v p 1ζ (m

V s

)

0.5 1 1.5 2 2.50.095

0.48

3.0

5 6 7 8 9

0

2

4

6

φ e (H

z)

5 6 7 8 9

−5

5

15

φ e (H

z)

5 6 7 8 9

−20

10

50

Time (s)

φ e (H

z)

0

0.5

1

1.5

2

2.5

3

A

IV

II

I

B

III

C1

C2

vp

1ζ = 3.0mV s

vp

1ζ = 0.09mV s

C3

vp

1ζ = 0.6mV s

C

Figure 7: An example illustrating that the bidirectional control of absence seizures by the

basal ganglia is also available for the corticothalamic loop transmission delay induced SWDs.

A, B: Tow-dimensional state analysis (A) and corresponding frequency analysis (B) in the (K, vp1ζ) panel.

Here K is the scale factor, and vp1ζ represents the excitatory coupling strength of the STN-SNr pathway.

Four different dynamical states are observed in (A): the saturation state (I), the SWD oscillation state (II),

the simple oscillation state (III) and the low firing state (IV), which also correspond to previous figures. In

(A), the region between two red dashed lines denotes the main bidirectional suppression region of SWDs,

where the double arrow represents both increasing and decreasing the excitatory coupling strength vp1ζ

can inhibit the SWDs. In (B), the asterisk (“∗”) region surrounded by dashed lines denotes the typical

2-4 Hz SWD oscillation region. (C) Three typical time series of φe for different values of vp1ζ , with

K = 1.3. Here we choose vp1ζ = 0.09 mV s (top), vp1ζ = 0.6 mV s (middle) and vp1ζ = 3.0 mV s

(bottom), respectively. In all simulations, we set t0 = 80 ms, ves = 3.2 mV s, vse = 3.4 mV s, vre = 1.6

mV s, vsr = −1.76 mV s and φn = 8.0 mV s.

19

here we block the GABAB pathway from TRN to SRN, and consider a bidirectional t0/2 transmission

delay between the cerebral cortex and thalamus, as that used in previous studies [11, 33]. Additionally,

several coupling strengths within the corticothalamic loop are also needed to be adapted, because such

transmission delay induced SWDs require strong interactions between the cerebral cortex and thalamus

[11, 33]. The new added and modified model parameters that we used in this subsection are as follows

[11, 33]: t0 = 80 ms, ves = 3.2 mV s, vse = 3.4 mV s, vre = 1.6 mV s, vsr = −1.76 mV s and φn = 8.0

mV s. For simplicity, we term the current model as the modified model in the following studies.

Figures 7A and 7B show an example pair of state analysis and frequency analysis in the (K, vp1ζ)

panel, respectively. As expected, due to the competition between the SNr-TRN and SNr-SRN pathways,

we observe the significant bidirectional control feature for intermediate scale factor K (Fig. 7A, the region

between dashed lines). In this bidirectional region, both enhancing and lowing the excitatory coupling

strength vp1ζ push the model dynamics from the SWD oscillation state into the simple oscillation state

(Fig. 7C), thus inhibiting the generation of SWDs. This finding supports our above hypothesis that

under suitable conditions the basal ganglia may control and modulate the absence seizure activities

bidirectionally. However, compared to the results in Fig.5, we find that the bidirectional region appears

in a relatively larger K region for the modified model, and increasing the coupling strength vp1ζ cannot

kick the model dynamics into the low firing state as well. This is not so surprising because these two SWD

generation mechanisms that we used are similar but not completely identical. As we introduced above,

the SWDs induced by the corticothalamic loop transmission delay require relatively stronger interactions

between the cerebral cortex and thalamus, which essentially weaken the inhibitory effect from the SNr

neurons. This might lead that the firing of SRN neurons cannot be fully suppressed even when the

activation level of SNr reaches its saturation state. Another important finding that we discover here is

that the dynamics of the modified model become complicated for large K and strong vp1ζ (see Fig. 7A,

upper right). In additional simulations, we further perform a series of state analysis for the modified

model using the same group of parameter values but different random initial conditions (see Figure S3).

The corresponding results indicate that the modified model shows bistability (the simple oscillation state

or the saturation state) in the large K and strong vp1ζ region, and final model dynamics significantly

depend on the initial conditions.

In conclusion, these findings further stress the combination role of the inhibitory SNr-TRN and

SNr-SRN pathways on the control of absence seizure activities. By combining all of our results, we

postulate that the bidirectional control by the basal ganglia is possible a generalized regulatory mechanism

for absence seizures and may be extendable to other pathological factors, even though the detailed

bidirectional control behaviors may not be completely identical for different pathological factors.

Discussion

Using a mean-field macroscopic model that incorporates the basal ganglia, cerebral cortex and tha-

lamus, we presented here the first investigation on how the basal ganglia control the absence epilepsy

through the projections directly emitted from the SNr to several key nuclei of thalamus. Through sim-

ulations, we demonstrated that the absence seizure activities induced by the slow synaptic kinetics of

GABAB in TRN can be inhibited by either the isolated SNr-TRN pathway or the isolated SNr-SRN

pathway via different biophysical mechanisms. More importantly, our results showed that under certain

conditions these two types of seizure control can coexist in the same network, suggesting that both de-

creasing and increasing the activation levels of SNr may considerably suppress the generation of SWDs.

Theoretically, such bidirectional control of absence seizures by basal ganglia is due to the effective com-

petition between the SNr-TRN and SNr-SRN pathways, which might be a generalized mechanism for

20

regulating absence seizure activities and can be extended to other pathological factors. In addition, our

detailed frequency analysis also indicated that, depending on different system conditions, the developed

model may exist low, high or both TMFRs for the SNr neural population for triggering the typical 2-4

Hz SWDs.

These results are at least partly in agreement with former experimental observations. Previously,

experimental studies based on electrophysiological recordings have established the linkage that reducing

the activation of SNr neurons from the normal level can effectively suppress the SWDs in different rodent

animal models [21–24]. Such antiepileptic effect was supposed to be attributed to the indirect pathway of

the SNr to TRN relaying at superior colliculus. The results presented in this work also demonstrated that

decreasing the SNr activity is an effective approach that terminates the SWDs. However, it is important

to note that in our model the similar antiepileptic effect is triggered by the direct GABAergic projections

from SNr to TRN. Presumably, this is because the SNr has overall inhibitory impacts on TRN via both

indirect and direct pathways. In the brain of absence epileptic patients, both of these two pathways might

work together and provide a stable and endogenous mechanism to terminate the paroxysm of absence

epilepsy.

Our model further makes prediction that increasing the activation of SNr from the normal level may

also suppress SWDs. In previous experimental studies, there still lacks sufficient evidence to support this

viewpoint. We speculate that this might be because the relative strength of the SNr-TRN and SNr-SRN

pathways for rodent animals is generally high or at least not too low. Under this condition, the BGCT

system for most rodent animals does not operate in the bidirectional control region, thus the suppression

of SWDs caused by activating SNr is difficult to be observed in normal experiments. According to our

results, the activation of SNr induced SWD suppression might appear by suitably tuning the relative

strength between these two pathways. Further experiments based on animal models will be necessary to

validate this prediction and characterize the detailed nature of the SWD suppression induced by the SNr-

SRN pathway. Even so, the above prediction from our computational study might provide an alternative

approach for terminating absence seizures.

An interesting and important question is: can real brain utilize this bidirectional modulation mecha-

nism to control the paroxysm of absence epilepsy? Our answer is that it is possible, if the real brain has

some mechanisms to automatically adjust the balance between the SNr-TRN and SNr-SRN pathways.

Theoretically, there are several possible biological mechanisms and one of them is discussed as follows.

Experimental data have uncovered that synapses conduct signals in an unreliable fashion, which is due to

the probabilistic neurotransmitter release of synaptic vesicles [44–46]. It has been shown that the trans-

mission failure rate at a given synapse generally tends to exceed the fraction of successful transmission,

and in some specific cases it can be even higher than 0.9 [44–46]. Interestingly, recent studies indicated

that such synaptic unreliability may play critical functional roles in neural processing and computation

[47–49]. For patients with absence seizures, the suitable competition between the SNr-TRN and SNr-

SRN pathways can be theoretically achieved by properly tuning the synaptic transmission rate of these

pathways. This might be an important underlying mechanism and has significant functional advantages,

because it does not require the changes of related anatomical structures and connection densities in the

brain. However, we should notice that such tuning is not easy. Specifically, it requires the cooperation

among neurons and needs to change the synaptic transmission rates collectively for most of relevant

synapses. From the functional perspective, this mechanism may be associated with the self-protection

ability of brain. After a long time of evolution, it is reasonable to suppose that our brain might have

the abilities to use this type of plasticity-like mechanism to achieve complicated self-protection function

during absence seizures [49]. Nevertheless, additional well-designed experiments are still needed to test

whether our proposed hypothesis is correct.

21

The current results highlight the functional roles of basal ganglia in the control of absence seizures

and might offer physiological implications on different aspects. First, the termination of absence seizures

by activating SNr neurons through the SNr-SRN pathway might inspire the treatment of refractory

absence seizures. Although the absence epilepsy is one typical benign epilepsy, a considerable proportion

of patients yet may fail to achieve freedom from absence seizures and become refractory to multiple

antiepileptic drugs [50, 51]. For those patients, one possible reason is that the strengths of their direct and

indirect SNr-TRN pathways are somewhat weak, compared to the other patients. In this case, reducing

the activation level of SNr may lose efficacy in the suppression of SWDs and, contrarily, increasing the

firing of SNr neurons may stop the absence seizures. Second, our results might be generalized to the

GPi neurons. In addition to the SNr, the GPi is also an important output structure of basal ganglia.

The SNr often works in unison with GPi, since they have closely related inputs and outputs, as well as

similarities in cytology and function [30–32]. It is thus reasonable to infer that the activation level of GPi

might serve a similar bidirectional role in modulating the absence seizures. Third, the results presented

in this study may provide new insight into the deep brain stimulation therapy on absence seizures [52].

In previous studies, it has been demonstrated that absence seizures can be inhibited by suitably applying

the deep brain stimulation to STN. Our results not only support the traditional viewpoint, but also

indicate that the SNr may be a more direct therapeutic target, compared to STN. We thus infer that

the inhibition of typical absence seizure activities can also be accomplished by appropriately stimulating

the SNr neurons. Finally, the main goal of this study was to explore the control role of basal ganglia in

absence seizures, but not limited to this specific epilepsy. Several other types of epilepsies, such as the

juvenile myoclonic epilepsy [53] and the generalized tonic-clonic epilepsy [54], are also highly associated

with the corticothalamic system and mediated by GABA receptors. If our above findings on absence

seizures capture the real fact, such bidirectional control feature by basal ganglia may be also available

for these types of epilepsies.

Although our developed model is powerful enough to suggest some functional roles of basal ganglia in

controlling and modulating absence seizures, we have to admit that this biophysical model is simplified

and idealized. The limitations of this model and possible extensions in future studies need to be discussed.

First, more detailed models based on the spiking neurons are able to introduce much more complexities,

such as ionic dynamics, firing variability and connection property, to the system. In previous studies,

it has been observed that ion concentrations inside the cell and in the extracellular space as well as the

firing dynamics of neurons are changed during epileptic seizures [55–57]. It has been also found recently

that the connection property of neural systems in different scales (for example, neuronal networks at

microscopic scales and brain region networks at macroscopic scales) also contribute to the generation of

seizure-like activities [58, 59]. Therefore, we cannot exclude other possible interesting results by using

more detailed modelling methodologies. Moreover, due to lack of necessary data, our model does not

include the indirect pathway from the SNr to TRN, relaying at the intermediate and superficial layers of

the caudal superior colliculus [60]. Although we have inferred that both the direct and indirect SNr-TRN

pathways play the similar effects on the control of absence seizures, further studies based on detailed

anatomical data need to be investigated in the future work. Finally, it should be emphasized that in

the present study we mainly focus on the SWDs induced by the slow kinetics of GABAB receptors in

TRN. Even though we also showed that the similar bidirectional control behavior can be observed for

another pathological factor (i.e., the corticothalamic loop transmission delay), more detailed analysis on

this aspect is still needed. As we know, several other pathological factors, such as the increased T-type

Ca2+ current in thalamocortical neurons [2], may also lead to the generation of SWDs. Nevertheless, it

is still unknown and deserves to be further examined whether our proposed bidirectional control by basal

ganglia is also available for these other pathological factors.

22

Theoretically, a better choice for mimicking the slower dynamics of GABAB currents should use

relatively smaller values of α and β at the relevant synapses. However, we have to notice that the mean-

filed theory used in the present study is under an assumption of uniform α and β parameters for all

incoming connections of one neural population (see Eq. 3). To our knowledge, introducing different α

and β to different types of connections will make such simple form of differential operator presented in

Eq. 3 become invalid, thus making the dendritic filtering of incoming signals mathematically intractable

in the current mean-field framework. Considering this, we employed a delay parameter τ to represent

the order of magnitude of rise and decay time of GABAB currents in our model, as the method used

in a recent neural mean-field modelling study [13]. Essentially, more detailed models based on spiking

neurons will allow us to consider different dynamical processes for different types of synapses [49, 61].

We are currently trying to extend our above-identified results to a relevant spiking neural network and

the corresponding results will be presented elsewhere.

It should be also pointed out that traditional modelling studies on absence seizures mainly focus

on the possible generation mechanism of the typical absence seizure activities, i.e., SWDs, within the

corticothalamic system, but little research addresses the regulatory mechanisms of absence seizures by

basal ganglia. Inspired by previous experimental observations [21–24] and recent modelling studies on

Parkinson’s disease [30, 31], here we introduced the basal ganglia to the traditional mean-field model of

corticothalamic system proposed by Robinson and his collaborators, and firstly explored how the basal

ganglia control the SWDs through the computational approach. Theoretically, several other models that

describe the macro-scale dynamics about the activity of neural ensembles, such as the Wilson-Cowan

model [62–64] and the models based on dynamic causal modelling [65–67], can be also used to deal with

the similar problem but may need to re-estimate parameters according to different model assumptions.

In summary, we have performed mechanistic studies and investigated the detailed roles of basal ganglia

in the control of absence seizures. We have computationally demonstrated that the SWDs generated by

the developed model can be terminated and modulated by both decreasing and increasing the activation

levels of SNr. Our results provide the first evidence on the bidirectional control of absence seizures by

basal ganglia. This finding might deepen our conventional understanding about the functional roles of

basal ganglia in the controlling and modulating of absence seizures. For patients with absence epilepsy,

these results in turn indicate that the loss of several basal ganglia functions, especially the functions

related to the SNr neurons, might further aggravate the attacks of absence epilepsy. We hope that

predictions from our systematic model investigation can not only inspire testable hypotheses for future

electrophysiological experiments but also provide additional therapeutic strategies for absence seizures.

Acknowledgements

We sincerely thank Dr. Dan Wu and Dr. Yangsong Zhang for valuable comments on early versions of

this manuscript. This work is supposed by the 973 project (No. 2011CB707803), the National Natural Sci-

ence Foundation of China (No. 61201278, No. 81071222, No. 81371636, No. 81330032, and No. 91232725),

and the 111 project (B12027).

Author Contributions

Conceived and designed the experiments: MC DG TW YX PX PAVS DY. Performed the experiments:

MC DG CL. Analyzed the data: MC DG TW WJ YX DY. Wrote the paper: MC DG YX CL PAVS DY.

23

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27

Supporting Information

Figure S1: Several other typical time series of φe (left frames) and their corresponding

spectra (right frames) generated by our developed BGCT model. To a certain extent, these

model time series are comparable with real physiological EEG signals: eyes-open (A) and (B), alpha

rhythm (C), beta rhythm (D), coexistence of alpha and beta rhythms (E), and polyspike and wave (F).

Here we adjust the excitatory corticothalamic coupling strength vse and delay parameter τ for reproducing

these time series. The detailed parameter values used in our simulations are: vse = 0.6 mV s and

τ = 32 ms (A), vse = 1.1 mV s and τ = 16 ms (B), vse = 1.2 mV s and τ = 16 ms (C), vse = 1.25 mV s

and τ = 3 ms (D), vse = 1.22 mV s and τ = 16 ms (E), and vse = 2.2 mV s and τ = 70 ms (F),

respectively. In addition, we also introduce a certain level of gaussian white noise into φn, with the mean

〈φn〉=2.0 mV s and standard deviation σ(φn) = 0.0141 mV s1/2. Note that more types of model time

series might be observed by further tuning these critical parameter values.

Figure S2: An example representation of the bidirectional control of absence seizures by

the basal ganglia in a relatively larger scale factor interval. The state analysis (A) and frequency

analysis (B) in the (K, vp1ζ) panel, with the inhibitory coupling strength vsr = −1.2 mV s. Here K is

the scale factor, and vp1ζ is the excitatory coupling strength of the STN-SNr pathway used to control the

activation level of SNr neurons. Similar to the results in Figs. 5A and 6A, only three dynamical state

regions are observed in the phase diagram (A): the SWD oscillation region (II), the simple oscillation

region (III) and the low firing region (IV). In (A), the region marked by diamond denotes the whole

suppression regions of SWDs, the white dashed line represents the boundary of suppression region, and

the red dashed line stands for the demarcation between the bidirectional (double arrow) and unidirectional

(single arrow) suppression regions. In (B), the asterisk (“∗”) region surrounded by dashed lines denotes

the typical 2-4 Hz SWD oscillation region. Compared to the results shown in Figs. 5 and 6, here we

consider a relatively larger scale factor interval from 0 to 2.3.

Figure S3: A series of two-dimensional state analysis in the (K, vp1ζ) panel for the modified

model. In (A)-(H), we use the same group of parameter values but different random initial conditions

for simulations. Similar to previous results, four different dynamical states are observed: the saturation

state (I), the SWD oscillation state (II), the simple oscillation state (III) and the low firing state (IV).

In each subfigure, the region between two red dashed lines denotes the main bidirectional suppression

region of SWDs, where the double arrow represents that both increasing and decreasing vp1ζ can inhibit

the generation of SWDs. The results given in (A)-(H) indicate that the modified model shows bistability

(the simple oscillation state or the saturation state) in the large K and strong vp1ζ region, and the

final dynamics of the modified model significantly depend on the initial conditions. Note that in all

simulations, we set t0 = 80 ms, ves = 3.2 mV s, vse = 3.4 mV s, vre=1.6 mV s, vsr = −1.76 mV s and

φn = 8.0 mV s.

Text S1: Supporting information code.

Text S2: Supporting information code.

28

11 12 13 14 15 16

17

18

19

φ e (H

z)

11 12 13 14 15 16

19

19.5

20

20.5

φ e (H

z)

11 12 13 14 15 16

14

16

18

20

22

φ e (H

z)

11 12 13 14 15 16

0

100

200

φ e (H

z)

11 12 13 14 15 16

12

12.5

13

φ e (H

z)

100

101

102

100

P(f

)

100

101

102

10−5

P(f

)10

010

110

2

10−5

P(f

)

100

101

102

100

P(f

)

100

101

102

100

P(f

)

11 12 13 14 15 16

3

3.2

3.4

φ e (H

z)

100

101

102

10−5

P(f

)

Time ( s ) Frequency ( Hz )

8.789 Hz

9.766 Hz

15.63 Hz

A A

1 A

2

B B

1 B

2

C C

2

D

E

F

D2

E2

F2

C1

D1

E1

F1

18.55 Hz

[Figure S1 legend]

29

Dominant frequency (Hz)

v p 1ζ (m

V s

)

Scale factor K0 1.15 2.3

0.03

0.3

3

0

1

2

3

4

5

6

0 1.15 2.30.03

0.3

3State region

Scale factor K

v p 1ζ (m

V s

)

IIIIV

II

A B

[Figure S2 legend]

30

v p1ζ (

mV

s)

0.5 1 1.5 2 2.50.095

0.48

3.0

Scale factor K

v p1ζ (

mV

s)

0.5 1 1.5 2 2.50.095

0.48

3.00.5 1 1.5 2 2.5

0.095

0.48

3.0

Scale factor K0.5 1 1.5 2 2.5

0.095

0.48

3.00.5 1 1.5 2 2.5

0.095

0.48

3.0

Scale factor K0.5 1 1.5 2 2.5

0.095

0.48

3.00.5 1 1.5 2 2.5

0.095

0.48

3.0

Scale factor K

0.5 1 1.5 2 2.50.095

0.48

3.0

(Hz)

IVIIIIII

A B C D

HGFE

[Figure S3 legend]

31